行列の計算例題（行列式（余因子展開））

　　　　　　　　　　


例題：次の行列式を余因子展開を使って計算せよ。

\[ \left|\,\begin{array}{@{}rwr{20pt}wr{20pt}wr{20pt}wr{20pt}@{}}{-}2 & 2 & {-}2 & {-}1 & 1 \\ {-}2 & 0 & {-}1 & 0 & 0 \\ 2 & 2 & 1 & {-}2 & 2 \\ {-}1 & {-}1 & {-}2 & 0 & {-}1 \\ 2 & {-}1 & 0 & 0 & 0\end{array}\,\right| \]

解答

\( \left|\,\begin{array}{@{}rwr{15pt}wr{15pt}wr{15pt}wr{15pt}@{}}{-}2 & 2 & {-}2 & {-}1 & 1 \\ {-}2 & 0 & {-}1 & 0 & 0 \\ 2 & 2 & 1 & {-}2 & 2 \\ {-}1 & {-}1 & {-}2 & 0 & {-}1 \\ 2 & {-}1 & 0 & 0 & 0\end{array}\,\right| \qquad \) （余因子展開） 第5列で展開する : [1, 0, 2, -1, 0]

\( \qquad\quad = (-1)^{5-1}\times1\times\left|\,\begin{array}{@{}rwr{15pt}wr{15pt}wr{15pt}@{}}{-}2 & 0 & {-}1 & 0 \\ 2 & 2 & 1 & {-}2 \\ {-}1 & {-}1 & {-}2 & 0 \\ 2 & {-}1 & 0 & 0\end{array}\,\right| + (-1)^{5-3}\times2\times\left|\,\begin{array}{@{}rwr{15pt}wr{15pt}wr{15pt}@{}}{-}2 & 2 & {-}2 & {-}1 \\ {-}2 & 0 & {-}1 & 0 \\ {-}1 & {-}1 & {-}2 & 0 \\ 2 & {-}1 & 0 & 0\end{array}\,\right| + (-1)^{5-4}\times(-1)\times\left|\,\begin{array}{@{}rwr{15pt}wr{15pt}wr{15pt}@{}}{-}2 & 2 & {-}2 & {-}1 \\ {-}2 & 0 & {-}1 & 0 \\ 2 & 2 & 1 & {-}2 \\ 2 & {-}1 & 0 & 0\end{array}\,\right| \)

\( \qquad\quad = (-1)^{5-1}\times1\times(-2) + (-1)^{5-3}\times2\times1 + (-1)^{5-4}\times(-1)\times(-8) \)

\( \qquad\quad = -8 \)

\( \left|\,\begin{array}{@{}rwr{15pt}wr{15pt}wr{15pt}@{}}{-}2 & 0 & {-}1 & 0 \\ 2 & 2 & 1 & {-}2 \\ {-}1 & {-}1 & {-}2 & 0 \\ 2 & {-}1 & 0 & 0\end{array}\,\right| \qquad \) （余因子展開） 第4列で展開する : [0, -2, 0, 0]

\( \qquad\quad = (-1)^{4-2}\times(-2)\times\left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}{-}2 & 0 & {-}1 \\ {-}1 & {-}1 & {-}2 \\ 2 & {-}1 & 0\end{array}\,\right| \)

\( \qquad\quad = (-1)^{4-2}\times(-2)\times1 \)

\( \qquad\quad = -2 \)

\( \left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}{-}2 & 0 & {-}1 \\ {-}1 & {-}1 & {-}2 \\ 2 & {-}1 & 0\end{array}\,\right| \qquad \) （余因子展開） 第3列で展開する : [-1, -2, 0]

\( \qquad\quad = (-1)^{3-1}\times(-1)\times\left|\,\begin{array}{@{}rwr{15pt}@{}}{-}1 & {-}1 \\ 2 & {-}1\end{array}\,\right| + (-1)^{3-2}\times(-2)\times\left|\,\begin{array}{@{}rwr{15pt}@{}}{-}2 & 0 \\ 2 & {-}1\end{array}\,\right| \)

\( \qquad\quad = (-1)^{3-1}\times(-1)\times3 + (-1)^{3-2}\times(-2)\times2 \)

\( \qquad\quad = 1 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}{-}1 & {-}1 \\ 2 & {-}1\end{array}\,\right| = (-1)\times(-1) - (-1)\times2 = 3 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}{-}2 & 0 \\ 2 & {-}1\end{array}\,\right| = (-2)\times(-1) - 0\times2 = 2 \)

\( \left|\,\begin{array}{@{}rwr{15pt}wr{15pt}wr{15pt}@{}}{-}2 & 2 & {-}2 & {-}1 \\ {-}2 & 0 & {-}1 & 0 \\ {-}1 & {-}1 & {-}2 & 0 \\ 2 & {-}1 & 0 & 0\end{array}\,\right| \qquad \) （余因子展開） 第4列で展開する : [-1, 0, 0, 0]

\( \qquad\quad = (-1)^{4-1}\times(-1)\times\left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}{-}2 & 0 & {-}1 \\ {-}1 & {-}1 & {-}2 \\ 2 & {-}1 & 0\end{array}\,\right| \)

\( \qquad\quad = (-1)^{4-1}\times(-1)\times1 \)

\( \qquad\quad = 1 \)

\( \left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}{-}2 & 0 & {-}1 \\ {-}1 & {-}1 & {-}2 \\ 2 & {-}1 & 0\end{array}\,\right| \qquad \) （余因子展開） 第3列で展開する : [-1, -2, 0]

\( \qquad\quad = (-1)^{3-1}\times(-1)\times\left|\,\begin{array}{@{}rwr{15pt}@{}}{-}1 & {-}1 \\ 2 & {-}1\end{array}\,\right| + (-1)^{3-2}\times(-2)\times\left|\,\begin{array}{@{}rwr{15pt}@{}}{-}2 & 0 \\ 2 & {-}1\end{array}\,\right| \)

\( \qquad\quad = (-1)^{3-1}\times(-1)\times3 + (-1)^{3-2}\times(-2)\times2 \)

\( \qquad\quad = 1 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}{-}1 & {-}1 \\ 2 & {-}1\end{array}\,\right| = (-1)\times(-1) - (-1)\times2 = 3 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}{-}2 & 0 \\ 2 & {-}1\end{array}\,\right| = (-2)\times(-1) - 0\times2 = 2 \)

\( \left|\,\begin{array}{@{}rwr{15pt}wr{15pt}wr{15pt}@{}}{-}2 & 2 & {-}2 & {-}1 \\ {-}2 & 0 & {-}1 & 0 \\ 2 & 2 & 1 & {-}2 \\ 2 & {-}1 & 0 & 0\end{array}\,\right| \qquad \) （余因子展開） 第4列で展開する : [-1, 0, -2, 0]

\( \qquad\quad = (-1)^{4-1}\times(-1)\times\left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}{-}2 & 0 & {-}1 \\ 2 & 2 & 1 \\ 2 & {-}1 & 0\end{array}\,\right| + (-1)^{4-3}\times(-2)\times\left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}{-}2 & 2 & {-}2 \\ {-}2 & 0 & {-}1 \\ 2 & {-}1 & 0\end{array}\,\right| \)

\( \qquad\quad = (-1)^{4-1}\times(-1)\times4 + (-1)^{4-3}\times(-2)\times(-6) \)

\( \qquad\quad = -8 \)

\( \left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}{-}2 & 0 & {-}1 \\ 2 & 2 & 1 \\ 2 & {-}1 & 0\end{array}\,\right| \qquad \) （余因子展開） 第3列で展開する : [-1, 1, 0]

\( \qquad\quad = (-1)^{3-1}\times(-1)\times\left|\,\begin{array}{@{}rwr{15pt}@{}}2 & 2 \\ 2 & {-}1\end{array}\,\right| + (-1)^{3-2}\times1\times\left|\,\begin{array}{@{}rwr{15pt}@{}}{-}2 & 0 \\ 2 & {-}1\end{array}\,\right| \)

\( \qquad\quad = (-1)^{3-1}\times(-1)\times(-6) + (-1)^{3-2}\times1\times2 \)

\( \qquad\quad = 4 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}2 & 2 \\ 2 & {-}1\end{array}\,\right| = 2\times(-1) - 2\times2 = -6 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}{-}2 & 0 \\ 2 & {-}1\end{array}\,\right| = (-2)\times(-1) - 0\times2 = 2 \)

\( \left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}{-}2 & 2 & {-}2 \\ {-}2 & 0 & {-}1 \\ 2 & {-}1 & 0\end{array}\,\right| \qquad \) （余因子展開） 第3列で展開する : [-2, -1, 0]

\( \qquad\quad = (-1)^{3-1}\times(-2)\times\left|\,\begin{array}{@{}rwr{15pt}@{}}{-}2 & 0 \\ 2 & {-}1\end{array}\,\right| + (-1)^{3-2}\times(-1)\times\left|\,\begin{array}{@{}rwr{15pt}@{}}{-}2 & 2 \\ 2 & {-}1\end{array}\,\right| \)

\( \qquad\quad = (-1)^{3-1}\times(-2)\times2 + (-1)^{3-2}\times(-1)\times(-2) \)

\( \qquad\quad = -6 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}{-}2 & 0 \\ 2 & {-}1\end{array}\,\right| = (-2)\times(-1) - 0\times2 = 2 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}{-}2 & 2 \\ 2 & {-}1\end{array}\,\right| = (-2)\times(-1) - 2\times2 = -2 \)